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| Also see {{#ask: [[fact about::normal subgroup]]|limit = 0|searchlabel = facts about normality}}, {{#ask: [[proves property satisfaction of::normal subgroup]]|limit = 0|searchlabel = facts that prove that a subgroup is normal}}, and {{#ask:[[uses property satisfaction of::normal subgroup]]|limit = 0|searchlabel = facts that use that a subgroup is normal}}. | | Also see {{#ask: [[fact about::normal subgroup]]|limit = 0|searchlabel = facts about normality}}, {{#ask: [[proves property satisfaction of::normal subgroup]]|limit = 0|searchlabel = facts that prove that a subgroup is normal}}, and {{#ask:[[uses property satisfaction of::normal subgroup]]|limit = 0|searchlabel = facts that use that a subgroup is normal}}. |
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| Q: '''There are so many different definitions of normal subgroup. Which of these is the ''correct'' definition? Which one should I use in practical situations?'''
| | {{#ask: [[question about::normal subgroup]]|?question about|?question type|?answer references}} |
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| A: The two most important definitions of normality are the definition as the [[kernel]] of a [[homomorphism of groups]] and the definition in terms of invariance under [[inner automorphism]]s. However, the other definitions are also important and worth knowing. What's really important about normality is not any one of these definitions ''per se'', but the remarkable fact that all these definitions are equivalent. To see how different definitions can be used, both for proving that a subgroup is normal and for using that a subgroup is normal, see [[proving normality]] and [[using normality]].
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| Q: '''The definition of normality seems to be somewhat related to the notion of [[abelian group]]. What precisely is the relation?'''
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| A: The main relation is [[abelian implies every subgroup is normal]]: in an abelian group, every subgroup is normal. This is best seen from the conjugation/inner automorphism definition, because in an abelian group, every element is invariant under conjugation by every other element. It can also be seen from the left coset/right coset definition or the commutator definition. Interestingly, there do exist non-abelian groups in which every subgroup is normal, such as the [[quaternion group]]. {{further|[[Dedekind not implies abelian]]}}
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| It is ''not'' necessary that an abelian subgroup of a non-abelian group be normal. The easiest counterexample is the subgroups of order two in [[symmetric group:S3|the symmetric group of degree three]] (see [[S2 is not normal in S3]]). It is ''also'' not true that any subgroup of an [[abelian normal subgroup]] is normal. An example is the [[dihedral group:D8]], which has an abelian normal subgroups of order four (the [[Klein four-subgroups of dihedral group:D8|Klein four-subgroups]]) which in turn have subgroups that are not normal in the whole group.
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| Q: '''I also learned something about the [[center]]. What is the difference between the notion of center and the notion of normal subgroup?'''
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| A: The center is defined as the set of elements that are, as individual elements, invariant under conjugation by other elements. On the other hand, a normal subgroup has to be invariant under conjugation as a set -- conjugation may move elements within the set.
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| It is true that the [[center is normal]]. More generally, a [[central subgroup]] is a subgroup of the center, and [[central implies normal|any central subgroup is normal]].
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| However, every normal subgroup need not be central. In fact, even an [[abelian normal subgroup]] need not be central. Examples include the normal subgroup of order three in the [[symmetric group:S3|symmetric group of degree three]] and the normal subgroups of order four in the [[dihedral group:D8]].
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| Q: '''One definition I saw said that <math>N</math> is normal in <math>G</math> if <math>gNg^{-1} \le N</math> for all <math>g \in G</math>. Another definition used <math>gNg^{-1} = N</math>. Why are these equivalent?'''
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| A: This is a very good and somewhat tricky question. The <math>=</math> formulation seems stronger, but it turns out to be equivalent. Interestingly, it is ''not'' true that if, for a particular element <math>g</math> and a subgroup <math>N</math>, <math>gNg^{-1} \le N</math>, then <math>gNg^{-1} = N</math>. Rather, it is the fact that this is true for ''every'' <math>g</math> that matters.
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| This is because [[restriction of automorphism to subgroup invariant under it and its inverse is automorphism]]. Now, if <math>N</math> is normal in <math>G</math> by the <math>\le</math> definition, we not only have <math>gNg^{-1} \le N</math>, we also have that <math>g^{-1}Ng \le N</math>. Thus, <math>N</math> is invariant both under conjugation by <math>g</math> and under conjugation by <math>g^{-1}</math>, which are inverse operations of each other. This forces <math>gNg^{-1} = N</math>. (To see this without using the fact quoted above, note that <math>g^{-1}Ng \le N</math>, implies, via left multiplication by <math>g</math> and right multiplication by <math>g^{-1}</math>, that <math>N \le gNg^{-1}</math>. Combined with <math>gNg^{-1} \le N</math>, we obtain <math>N = gNg^{-1}</math>).
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| Note also that the use of the <math>\le</math> symbol, as opposed to the <math>\subseteq</math> symbol, already encodes the information that conjugation by <math>g</math> is an automorphism, hence the image of <math>N</math> is a subgroup. If you read this definition before these ideas were introduced, you may have seen the notation <math>gNg^{-1} \subseteq N</math>. The same ideas apply with the set notation.
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| Q: '''I was told that it is very important and counterintuitive that a normal subgroup of a normal subgroup need not be normal, but I didn't find it either important or counterintuitive. What is the significance?'''
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| A: You're referring to the fact that [[normality is not transitive]]. This is indeed important, though it is not necessarily counterintuitive. The importance is partly because if the opposite were true, it would prove a very convenient way of showing that subgroups are normal, and thus make group theory very different and perhaps more boring.
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| One way of thinking about the significance is to look at the implications for quotient groups. If <math>H \le K \le G</math> with <math>H</math> normal in <math>K</math> and <math>K</math> normal in <math>G</math>, we can talk of the quotient groups <math>G/K</math> and <math>K/H</math>. If it were also true that <math>H</math> is normal in <math>G</math>, we'd have a group <math>G/H</math>, whereby <math>K/H</math> could be identified with a normal subgroup of it and the quotient would be isomorphic to <math>G/K</math> (by the [[third isomorphism theorem]]). However, the point is that we are ''not'' guaranteed that <math>H</math> is normal in <math>G</math>. This fact shows that there is, in some sense, a lot more flexibility in the way groups can be put together.
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| Q: '''I have seen the definition of normality as invariance under conjugations, but then read something about these conjugations also being called [[inner automorphism]]s. Does this have any significance?'''
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| A: That the conjugation operations are inner automorphisms, and that invariance under these is equivalent to being normal, is very significant. This is because inner automorphisms are automorphisms, so they preserve the group structure. This is one explanation for why [[normality is strongly join-closed]] (the subgroup generated by a bunch of normal subgroups is normal), [[normality is centralizer-closed]] (the centralizer of a normal subgroup is normal), and [[normality is commutator-closed]] (the commutator of two normal subgroups is normal). It also explains why [[characteristic implies normal|characteristic subgroups are normal]], which explains why [[subgroup-defining function]]s such as the [[center]], [[commutator subgroup]], [[socle]], and [[Frattini subgroup]] are all normal subgroups.
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| Q: '''Could you clarify the relationship between normal subgroups and characteristic subgroups?'''
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| A: Characteristic means invariant under all automorphisms, normal means invariant under inner automorphisms. [[Characteristic implies normal]], [[normal not implies characteristic]], and [[characteristic of normal implies normal]]. See more at [[characteristic versus normal]].
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| Q: '''I was told by some people that normal subgroups are precisely the subgroups that are invariant, but the correct term for that seems to be [[characteristic subgroup]]. How do you explain this?'''
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| A: It depends on what sort of invariance is being sought. If we are looking for invariance under [[automorphism]]s of the group, then the correct notion is that of [[characteristic subgroup]].
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| However, normality is more important in the following sense: when a group acts on a structure, we are interested in those subgroups of the group that are invariant under ''change of coordinates'' on the set, which means invariant under automorphisms of the set. Since the group itself acts by automorphisms, this in particular implies invariance under the action of the group. ''But when a group acts on a set, the induced action on itself is the group action by conjugation''. Thus, all subgroups invariant in this manner must be normal; however, they need not be characteristic. Since most of the applications of groups to other areas of mathematics is via group actions, normality is the more important notion. {{further|[[Ubiquity of normality]]}}
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| Q: '''The product of a normal subgroup with any subgroup is a subgroup. Are normal subgroups the only subgroups with this property?'''
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| A: No. A subgroup whose [[product of subgroups|product]] with any subgroup is a subgroup is termed a [[permutable subgroup]] (or quasinormal subgroup), i.e., it [[permuting subgroups|permutes]] with every subgroup. [[Normal implies permutable|normal subgroups are permutable]] but [[permutable not implies normal]]. For instance, any subgroup of the [[Baer norm]] (the intersection of normalizers of all subgroups) is permutable but it need not be normal.
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